High-ordered spectral characterization of unicyclic graphs

Abstract

In this paper we will apply the tensor and its traces to investigate the spectral characterization of unicyclic graphs. Let G be a graph and G m be the m-th power (hypergraph) of G. The spectrum of G is referring to its adjacency matrix, and the spectrum of G m is referring to its adjacency tensor. The graph G is called determined by high-ordered spectra (DHS, for short) if, whenever H is a graph such that H m is cospectral with G m for all m, then H is isomorphic to G. In this paper we first give formulas for the traces of the power of unicyclic graphs, and then provide some high-ordered cospectral invariants of unicyclic graphs. We prove that a class of unicyclic graphs with cospectral mates is DHS, and give two examples of infinitely many pairs of cospectral unicyclic graphs but with different high-ordered spectra.

Hypergraphs with Edge-Dependent Vertex Weights: Spectral Clustering Based on the 1-Laplacian

“…A cut is a partition of the vertex set V into two disjoint, nonempty subsets denoted by S and its complement S = V \ S. The weight of the cut is defined as the sum of cut costs associated with each hyperedge [13], i.e., cut(S, S) = e∈E we(S). The Cheeger constant [12,13,21] is defined as…”

“…Hence, different relaxations have been proposed, a popular one being spectral clustering, a relaxation based on the second eigenvector of the graph (or hypergraph) Laplacian [1,2,8]. This approach is also theoretically justified through the Cheeger inequality [6,12,13,21], where h2 is upper bounded by a function of the second smallest eigenvalue of the Laplacian. However, it has been proved that the Cheeger constant h2 is equal to the second smallest eigenvalue λ2 of the hypergraph 1-Laplacian, an alternative (non-linear) operator that generalizes the classical Laplacian (cf.…”

“…In this case, each hyperedge e is associated with a function we : 2 e → R ≥0 that assigns costs to every possible cut of the hyperedge where 2 e denotes the power set of e. The weight we(S) indicates the cost of partitioning e into the two subsets S and e \ S. In particular, when we is a submodular function, the corresponding model is called a submodular hypergraph which has desirable mathematical properties, making it convenient for theoretical analysis. A series of results in graph spectral theory [12] including p-Laplacians and Cheeger inequalities have been generalized to submodular hypergraphs [13,14].…”